I appreciate Dr. Herman Woltring's invitation to debate his proposal for
standardization of 3-D joint attitude representation. In this first response
I have three goals. These are: a) to outline my current position on Herman's
proposal; b) to address the common misconceptions about sequence dependency
of finite rotations, and c) to stimulate thought and discussion about the
description of particle displacements and how the principles used are applied
to the description of rigid body displacements.

My Current Position
Let me start with the area of agreement. I completely accept Dr. Woltring's
proposal for the use of the helical axis to describe joint attitude and
rotational displacement. The helical axis has the important characteristic
that the magnitude of the rotation and translation are independent of the
coordinate system chosen. The analogy to particle displacement is clear,
the magnitude of the total displacement vector is also independent of the
coordinate system chosen.

Now the area of disagreement. Herman has proposed the helical displacement
be decomposed into orthogonal components in either body segment's coordinate
system. This produces six components, three for translation and three for
rotation. I believe this is not fully satisfactory for describing joint
translation and incorrect for the joint rotation components. It is my
position that the proper joint rotation components are those describe by
Fred Suntay and I. My reasons for this will become apparent in the course of
the debate.

Misconceptions
Herman and many others refer to the rotations Fred and I described as being
an "ordered sequence of rotations". I would agree they are an ordered triple,
just like the orthogonal components of particle displacement are an ordered
triple. I disagree with the terminology "ordered sequence" because the
final displacement is not dependent upon the sequence the rotations are
performed. Am I missing some other meaning of this phrase?

There is a general, and incorrect, belief that finite three dimensional
rotations are sequence dependent. This is not surprising as almost every
text on the subject gives the example of a book rotated using two different
sequences resulting in two different final positions. This example is passed
along without any careful analysis of what is actually happening.

The basic problem with the book example is the three axes used are those of
an orthogonal coordinate system located in one of the body segments. While
such axes do define independent translational degrees-of-freedom (dof),
they do not define independent rotational dof. The independent rotational
degrees-of-freedom are those Herman referred to: a fixed axis in each body
segment and their common perpendicular. The orientation of the fixed axes
are chosen for convenience. This is similar to selecting an appropriate
orthogonal system for describing particle displacements.

To better understand the origin of the sequence dependency I will give a
similar example for particle displacement. It starts by first specifying
the displacements (x,y,z) without specifying the three independent dof.
Next, perform the displacements along the axes of any orthogonal
coordinate system and note its final location. Third change the orientation
of the orthogonal coordinate system. We still have three independent
dof but the directions have a different physical significance. Now
perform the three component displacements in any sequence. The final
position is clearly not the same as before. The problem is not that particle
displacements are sequence dependent, it's that we changed the independent
dof used.

Independent Rotational Degrees-of-Freedom
At the risk of being unnecessarily redundant I will again describe an
appropriate set of independent rotation axes. First locate two body fixed
axes, one in each body segment. These axes are selected so that rotation
about them is a motion of interest. The third rotation axis is the common
perpendicular to the two body fixed axes. The three angles which define the
orientation were described in the paper with Fred Suntay. Briefly, they
are:

1. The rotation about the common perpendicular axis is given by the
angle between the two body fixed axes.

2. The rotation about each body fixed axis is given by the angle
between the common perpendicular and a reference line located
in the same body as the fixed axis. It is convenient to
select the reference line so it is also perpendicular to the
body fixed axis. In this way the body fixed axes are normal
to the plane which contains both the reference line and the
common perpendicular axis.

In closing this first round of the debate I will state the primary reasons
for using the system we proposed as the components of the helical rotation.

1. They are independent components.

2. They add (in a screw sense) to the total helical rotation.

3. They correspond to common clinical descriptions of joint
rotation.

4. They are easy to compute from the rotation matrix and have a
well defined mathematical relationship with the total helical
rotation.